Abstract
We establish the structure of the dual Lie coalgebra for a Lie algebra of the symplectic Poisson bracket (Jacobian-type Poisson bracket) on the algebra of polynomials in evenly many variables. We show that if the base field has characteristic zero then the \( n \)-ary dual coalgebra for the Jacobian \( n \)-Lie algebra consists of the same linear functionals as the dual coalgebra for the commutative polynomial algebra.
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The work was supported by the RAS Fundamental Research Program (Project FWNF–2022–0002).
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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 5, pp. 992–1008. https://doi.org/10.33048/smzh.2023.64.508
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Zhelyabin, V.N., Kolesnikov, P.S. Dual Coalgebras of Jacobian \( n \)-Lie Algebras over Polynomial Rings. Sib Math J 64, 1153–1166 (2023). https://doi.org/10.1134/S0037446623050087
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DOI: https://doi.org/10.1134/S0037446623050087